To show that \( i \cdot j = j \cdot k = k \cdot i = 0 \), you must be referring to a context where these symbols (\( i \), \( j \), and \( k \)) represent elements of a specific mathematical system. One common context for this is the **quaternions**, where \( i \), \( j \), and \( k \) are imaginary units with specific multiplication rules.

In quaternions, the multiplication rules for \( i \), \( j \), and \( k \) are as follows:

\[
i^2 = j^2 = k^2 = ijk = -1
\]
Additionally, they satisfy the following properties:
\[
i \cdot j = k, \quad j \cdot k = i, \quad k \cdot i = j
\]
However, these products are not zero but rather yield another imaginary unit. If you mean something else or a different system, please clarify.

### If you are talking about a different mathematical system where \( i \), \( j \), and \( k \) are vectors or elements of a vector space or matrix, the result might be different, such as:

- **Cross product in vectors**: \( i \times j = k \), \( j \times k = i \), \( k \times i = j \)
- **Matrix elements** or **modular arithmetic** may also yield zero in certain cases.

Please clarify the specific system or provide more context. Is this related to quaternion algebra, vectors, or something else?

Would you like more details or have any questions?

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Here are five related questions:

1. What are quaternions and how do they generalize complex numbers?
2. How do the multiplication rules of \( i \), \( j \), and \( k \) in quaternions relate to rotations in 3D space?
3. Can you show how quaternion multiplication is associative but not commutative?
4. How do quaternions help in computer graphics, specifically in 3D rotations?
5. What are some other mathematical systems that include similar elements to \( i \), \( j \), and \( k \)?

**Tip**: When working with different algebraic structures, always check their defining properties (e.g., commutative, associative, or distributive laws).

Explore the properties of quaternions and learn why i * j, j * k, and k * i do not equal zero but instead yield another imaginary unit. This detailed explanation covers the fundamental multiplication rules and the non-commutative nature of quaternion algebra, suitable for Grades 11-12 and undergraduate students.

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